Final answer:
The horizontal component of the velocity of a projectile at its maximum height is constant, unaffected by gravity, and remains the same as its initial horizontal velocity since gravity only affects the vertical motion.
Step-by-step explanation:
The horizontal component of the velocity of a projectile object, when it has reached its maximum height, remains constant throughout the projectile's flight. This is because projectile motion is analyzed by breaking it into two independent motions: horizontal and vertical, as outlined in Figure 4.12(a).
For the horizontal motion, as per Figure 4.12(b), the acceleration in the horizontal direction (ax) is zero, which implies that the horizontal velocity (ux) is a constant and does not change due to gravity.
The maximum height of a projectile, as noted in the discussions, is determined solely by the vertical component of the initial velocity and the acceleration due to gravity. While the vertical component of the velocity becomes zero at maximum height, the horizontal component remains unchanged from its initial value.